Metric and Mixing Sufficient Conditions for Concentration of Measure

Details Metric and Mixing Sufficient Conditions for Concentration of Measure Metric and Mixing Sufficient Conditions for Concentration of Measure

2006-10-13

arxiv.org/abs/math/0610427v2

Mathematics

Probability

Keywords: concentration of measure, martingale differences, metric probability space, Levy family, strong mixing

Scientific paper

We derive sufficient conditions for a family $(X^n,\rho_n,P_n)$ of metric probability spaces to have the measure concentration property. Specifically, if the sequence $\{P_n\}$ of probability measures satisfies a strong mixing condition (which we call $\eta$-mixing) and the sequence of metrics $\{\rho_n\}$ is what we call $\Psi$-dominated, we show that $(X^n,\rho_n,P_n)$ is a normal Levy family. We establish these properties for some metric probability spaces, including the possibly novel $X=[0,1]$, $\rho_n=\ell_1$ case.

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